IndisputableMonolith.Relativity.Dynamics.RecognitionSheaf
The module defines the Recognition Sheaf as a sheaf of recognition potentials over the spacetime manifold, together with local sections, the J functional, stationarity conditions, and gluing axioms. Relativists working in Recognition Science cite it to equip dynamics with sheaf-theoretic structure that respects the J-cost and metric geometry. The module consists entirely of definitions and supporting declarations with no proofs.
claimLet $(M,g)$ be a spacetime manifold. The recognition sheaf $R$ is a sheaf whose sections are recognition potentials $f$ satisfying the $J$-cost functional, with local sections obeying stationarity at unity and global gluing via the recognition ratio.
background
The module imports Constants (where the RS time quantum satisfies $τ_0 = 1$ tick), Cost (which supplies the $J$-cost functional), and Metric (which equips spacetime with its geometry). It introduces the central object RecognitionSheaf together with LocalSection, the $J$ map, $J$-stationarity at one, section stationarity, local-to-global equality, recognition ratio unity, and sheaf gluing. These notions rest on the Recognition Composition Law and the $J$-uniqueness property from the upstream forcing chain.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the sheaf-theoretic layer for relativistic dynamics inside Recognition Science. It prepares the ground for theorems on section stationarity and gluing that connect to the eight-tick octave and $D=3$ spatial dimensions in the T0-T8 chain. Although no direct used-by edges are recorded yet, the definitions feed parent results on recognition potentials and local-global consistency.
scope and limits
- Does not prove existence or uniqueness of global sections.
- Does not derive equations of motion or field equations.
- Does not specify a concrete metric or coordinate chart.
- Does not incorporate mass formulas or the phi-ladder.
- Does not address Berry creation thresholds or alpha-band constraints.